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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoidl | Structured version Visualization version GIF version |
Description: A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
rngidl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
rngidl.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngoidl | ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3987 | . 2 ⊢ (𝑅 ∈ RingOps → 𝑋 ⊆ 𝑋) | |
2 | rngidl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
3 | rngidl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | eqid 2818 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
5 | 2, 3, 4 | rngo0cl 35078 | . 2 ⊢ (𝑅 ∈ RingOps → (GId‘𝐺) ∈ 𝑋) |
6 | 2, 3 | rngogcl 35071 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋) |
7 | 6 | 3expa 1110 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) ∈ 𝑋) |
8 | 7 | ralrimiva 3179 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋) |
9 | eqid 2818 | . . . . . . . . 9 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
10 | 2, 9, 3 | rngocl 35060 | . . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋) |
11 | 10 | 3com23 1118 | . . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋) |
12 | 2, 9, 3 | rngocl 35060 | . . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋) |
13 | 11, 12 | jca 512 | . . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
14 | 13 | 3expa 1110 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
15 | 14 | ralrimiva 3179 | . . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋)) |
16 | 8, 15 | jca 512 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))) |
17 | 16 | ralrimiva 3179 | . 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))) |
18 | 2, 9, 3, 4 | isidl 35173 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑋 ∈ (Idl‘𝑅) ↔ (𝑋 ⊆ 𝑋 ∧ (GId‘𝐺) ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑧(2nd ‘𝑅)𝑥) ∈ 𝑋 ∧ (𝑥(2nd ‘𝑅)𝑧) ∈ 𝑋))))) |
19 | 1, 5, 17, 18 | mpbir3and 1334 | 1 ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ran crn 5549 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 GIdcgi 28194 RingOpscrngo 35053 Idlcidl 35166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7103 df-ov 7148 df-1st 7678 df-2nd 7679 df-grpo 28197 df-gid 28198 df-ablo 28249 df-rngo 35054 df-idl 35169 |
This theorem is referenced by: divrngidl 35187 igenval 35220 igenidl 35222 |
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